We examine the relationship between three parameters of Type Ia supernovae (SNe~Ia): peak magnitude, rise time, and photospheric velocity at the time of peak brightness. The peak magnitude is corrected for extinction using an estimate determined from MLCS2k2 fitting. The rise time is measured from the well-observed $B$-band light curve with the first detection at least 1~mag fainter than the peak magnitude, and the photospheric velocity is measured from the strong absorption feature of Si~II~$\lambda$6355 at the time of peak brightness. We model the relationship among these three parameters using an expanding fireball with two assumptions: (a) the optical emission is approximately that of a blackbody, and (b) the photospheric temperatures of SNe~Ia are similar to each other at the time of peak brightness. We compare the precision of the distance residuals inferred using this physically motivated model against those from the empirical Phillips relation and the MLCS2k2 method for 47 low-redshift SNe~Ia ($0.005 < z< 0.04$) and find comparable scatter. However, SNe~Ia in our sample with higher velocities are inferred to be intrinsically fainter. Eliminating the high-velocity SNe and applying a more stringent extinction cut to obtain a “low-v-golden sample” of 22 SNe, we obtain significantly reduced scatter in the new relation, better than those of the Phillips relation and the MLCS2k2 method. After removing model peculiar velocities, our final scatter for the new relation is $0.108 \pm 0.018$~mag.